On the Use of Different Statistical Tests for Alert Correlation -- Short Paper
1.
On the use of diﬀerent statistical tests
for alert correlation – Short Paper
Federico Maggi and Stefano Zanero
Politecnico di Milano, Dip. Elettronica e Informazione
via Ponzio 34/5, 20133 Milano Italy
{fmaggi,zanero}@elet.polimi.it
Abstract. In this paper we analyze the use of diﬀerent types of sta-
tistical tests for the correlation of anomaly detection alerts. We show
that the Granger Causality Test, one of the few proposals that can be
extended to the anomaly detection domain, strongly depends on good
choices of a parameter which proves to be both sensitive and diﬃcult
to estimate. We propose a diﬀerent approach based on a set of simpler
statistical tests, and we prove that our criteria work well on a simpliﬁed
correlation task, without requiring complex conﬁguration parameters.
1 Introduction
One of the most challenging tasks in intrusion detection is to create a uniﬁed
vision of the events, fusing together alerts from heterogeneous monitoring de-
vices. This alert fusion process can be deﬁned as the correlation of aggregated
streams of alerts. Aggregation is the grouping of alerts that both are close in
time and have similar features; it fuses together diﬀerent “views” of the same
event. Alert correlation has to do with the recognition of logically linked alerts.
“Correlation” does not necessarily imply “statistical correlation”, but statistical
correlation based methods are sometimes used to reveal these relationships.
Alert fusion is more complex when taking into account anomaly detection
systems, because no information on the type or classiﬁcation of the observed
attack is available to the fusion algorithms. Most of the algorithms proposed
in the current literature on correlation make use of such information, and are
therefore inapplicable to purely anomaly based intrusion detection systems.
In this work, we explore the use of statistical causality tests, which have been
proposed for the correlation of IDS alerts, and which could be applied to anomaly
based IDS as well. We focus on the use of Granger Causality Test (GCT), and
show that its performance strongly depends on a good choice of a parameter
which proves to be sensitive and diﬃcult to estimate. We redeﬁne the causality
problem in terms of a simpler statistical test, and experimentally validate it.
2 Problem statement and state of the art
The desired output of an alert fusion process is a compact, high-level view of
what is happening on a (usually large and complex) network. In this work we use
2.
A1 A
.
. Normalization Prioritization Aggregation
.
An−1
A
An Visualization Verification Correlation
and management
Fig. 1. A diagram illustrating alert fusion terminology as used in this work.
a slightly modiﬁed version of the terminology proposed in [1]. Alerts streams are
collected from diﬀerent IDS sources, normalized and aggregated; alert correlation
is the very ﬁnal step of the process. In [1] the term “fusion” is used for the phase
we name “aggregation”, whereas we use the former to denote the whole process.
Fig. 1 summarizes the terminology.
In [2] we propose a fuzzy time-based aggregation technique, showing that it
yields good performance in terms of false positive reduction. Here, we focus on
the more challenging correlation phase. Eﬀective and generic correlation algo-
rithms are diﬃcult to design, especially if the objective is the reconstruction of
complex attack scenarios.
A technique for alert correlation based on state-transition graphs is shown
in [3]. The use of ﬁnite state automata enables for complex scenario descriptions,
but it requires known scenarios signatures. It is also unsuitable for pure anomaly
detectors which cannot diﬀerentiate among diﬀerent types of events. Similar
approaches, with similar strengths and shortcomings but diﬀerent formalisms,
have been tried with the speciﬁcation of pre- and post-conditions of the attacks
[4], sometimes along with time-distance criteria [5]. It is possible to mine scenario
rules directly from data, either in a supervised [6] or unsupervised [7] fashion.
Both approaches use alert classiﬁcations as part of their rules.
None of these techniques would work for anomaly detection systems, as they
rely on alert names or classiﬁcation to work. The best examples of algorithms
that do not require such features are based on time-series analysis and modeling.
For instance, [8] is based on the construction of time-series by counting the
number of alerts occurring into sampling intervals; the exploitation of trend
and periodicity removal algorithms allows to ﬁlter out predictable components,
leaving real alerts only as the output. More than a correlation approach, this is
a false-positive and noise-suppression approach, though.
The correlation approach investigated in [9] and based on the GCT also does
not require prior knowledge, and it drew our attention as one of the few viable
proposal for anomaly detection alert correlation in earlier literature. We will
describe and analyze this approach in detail in Section 4.
3 Problems in the evaluation of alert correlation systems
Evaluation techniques for alert fusion systems are still limited to a few proposals,
and practically and theoretically challenging to develop [2]. Additionally, the
common problem of the lack of reliable sources of data for benchmarking impacts
3.
heavily also on the evaluation of correlation systems. Ideally, we need both host
and network datasets, fully labeled, with complex attack scenarios described in
detail. These data should be freely available to the scientiﬁc community. These
requirements rule out real-world dumps.
The only datasets of this kind eﬀectively available are the ones by DARPA
(IDEVAL datasets). Of course, since this data set was created to evaluate IDS
sensors and not to assess correlation tools, it does not include sensor alerts. The
alerts have to be generated by running various sensors on the data. The 1999
dataset [10], which we used for this work, has many known shortcomings. Firstly,
it is evidently and hopelessly outdated. Moreover, a number of ﬂaws have been
detected and criticized in the network traces [11,12]. More recently, we analyzed
the host-based system call traces, and showed [13, 14] that they are ridden with
problems as well.
For this work these basic ﬂaws are not extremely dangerous, since the prop-
agation of attack eﬀects (from network to hosts) is not aﬀected by any of the
known ﬂaws of IDEVAL, and in fact we observed it to be quite realistically
present. What could be a problem is the fact that intrusion scenarios are too
simple and extremely straightforward. Additionally, many attacks are not de-
tectable in both network and host data (thus making the whole point of cor-
relation disappear). Nowadays, networks and attackers are more sophisticated
and attack scenarios are much more complex than in 1999, operating at various
layers of the network and application stack.
The work we analyze closely in the following [9] uses both the DEFCON 9
CTF dumps and the DARPA Cyber Panel Correlation Technology Validation
(CTV) [15] datasets for the evaluation of an alert correlation prototype. The
former dataset is not labeled and does not contain any background traﬃc, so in
fact (as the authors themselves recognize) it cannot be used for a proper evalua-
tion, but just for qualitative analysis. On the contrary, the DARPA CTV eﬀort,
carried out in 2002, created a complex testbed network, along with background
traﬃc and a set of attack scenarios. The alerts produced by various sensors dur-
ing these attacks were collected and given as an input to the evaluated correlation
tools. Unfortunately, this dataset is not available for further experimentation.
For all the previous reasons, in our testing we will use the IDEVAL dataset
with the following simpliﬁcation: we will just try to correlate the stream of
alerts coming from a single host-based IDS (HIDS) sensor with the correspond-
ing alerts from a single network-based IDS (NIDS), which is monitoring the
whole network. To this end, we ran two anomaly-based IDS prototypes (both
described in [13,14,16]) on the whole IDEVAL testing dataset. We ran the NIDS
prototype on tcpdump data and collected 128 alerts for attacks against the host
pascal.eyrie.af.mil [17]. The NIDS also generated 1009 alerts related to other
hosts. Using the HIDS prototype we generated 1070 alerts from the dumps of the
host pascal.eyrie.af.mil. With respect to these alerts, the NIDS was capable
of detecting almost 66% of the attacks with less than 0.03% of false positives;
the HIDS performs even better with a detection rate of 98% and 1.7% of false
positives.
4.
2.0
1.0
0.4
8
0.8
Granger Causality Index
Granger Causality Index
1.5
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p Value
p Value
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0 600 1200 1800 2400 0 600 1200 1800 2400 3500 6000 9000 12000 15000 50 100 150 200 250
Lag [seconds] Lag [seconds] Lag [seconds] Lag [seconds]
(1-a) (1-b) (2-a) (2-b)
Fig. 2. p-value (-a) and GCI (-b) vs. p with w = w1 = 60s (1-) and w = w2 = 1800s
(2-) “N etP (k) HostP (k)” (dashed line), “HostP (k) N etP (k)” (solid line).
In the following, we use this shorthand notation: N et is the substream of
all the alerts generated by the NIDS. HostP is the substream of all the alerts
generated by the HIDS installed on pascal.eyrie.af.mil, while N etP regards
all the alerts (with pascal as a target) generated by the NIDS; ﬁnally, N etO =
N etN etP indicates all the alerts (with all but pascal as a target) generated
by the NIDS.
4 The Granger Causality Test
In [9] Qin and Lee propose an interesting algorithm for alert correlation which
seems suitable also for anomaly-based alerts. Alerts with the same feature set
are grouped into collections of time-sorted items belonging to the same “type”
(following the concept of type of [8]). Subsequently, frequency time series are
built, using a ﬁxed size sliding-window: the result is a time-series for each collec-
tion of alerts. The prototype then exploits the GCT [18], a statistical hypothesis
test capable of discovering causality relationships between two time series when
they are originated by linear, stationary processes. The GCT gives a stochastic
measure, called Granger Causality Index (GCI), of how much of the history of
one time series (the supposed cause) is needed to “explain” the evolution of the
other one (the supposed consequence, or target). The GCT is based on the es-
timation of two models: the ﬁrst is an Auto Regressive model (AR), in which
future samples of the target are modeled as inﬂuenced only by past samples of
the target itself; the second is an Auto Regressive Moving Average eXogenous
(ARMAX) model, which also takes into account the supposed cause time series
as an exogenous component. A statistical F-test built upon the model estima-
tion errors selects the best-ﬁtting model: if the ARMAX ﬁts better, the cause
eﬀectively inﬂuences the target.
In [9] the unsupervised identiﬁcation of “causally related” events is performed
by repeating the above procedure for each couple of time-series. The advantage of
the approach is that it does not require prior knowledge (even if it may use attack
probability values, if available, for an optional prioritization phase). However,
in a previous work [2] we showed that the GCT fails however in recognizing
“meaningful” relationships between IDEVAL attacks.
We tested the sensitivity of the GCT to the choice of two parameters: the
sampling time, w, and the time lag p (that is, the order of the AR). In our simple
5.
10
Optimal p
6
2
0 2 4 6 8 10 12 14
Data range (10^3)
Fig. 3. The optimal time lag p given by the AIC criterion strongly varies over time.
ˆ
experiment, the expected result is that N etP HostP , and that HostP
N etP (the indicates “causality” while is its negation). In [9] the sampling
time was arbitrarily set to w = 60s, while the choice of p is not documented.
However, our experiments show that the choice of these parameters can strongly
inﬂuence the results of the test. In Fig. 2 (1-a/b) we plotted the p-value and
the GCI of the test for diﬀerent values of p (w = 60s). In particular, the dashed
line corresponds to the test N etP (k) HostP (k), and the solid line to the test
HostP (k) N etP (k). We recall that if the p-value is lower than the signiﬁcance
level, the null hypothesis is refused. Notice how diﬀerent choices of p can lead to
inconclusive or even opposite results. For instance, with α = 0.20 and with 2 ≤
p ≤ 3, the result is that N etP (k) HostP (k) and that HostP (k) N etP (k).
As we detailed in [2] (Fig. 2 (2-a/b)), other values of p lead to awkward result
that both HostP (k) N etP (k) and N etP (k) HostP (k).
A possible explanation is that the GCT is signiﬁcant only if both the linear
regression models are optimal, in order to calculate the correct residuals. If we
use the Akaike Information Criterion (AIC) [19] to estimate the optimal time
lag p over diﬀerent windows of data, we ﬁnd out that p wildly varies over time,
ˆ ˆ
as it is shown in Fig. 3. The fact that there is no stable optimal choice of p,
combined with the fact that the test result signiﬁcantly depends on it, makes
us doubt that the Granger causality test is a viable option for general alert
correlation. The choice of w seems equally important and even more diﬃcult to
perform, except by guessing.
Of course, our testing is not conclusive: the IDEVAL alert sets may simply
not be adequate for showing causal relationships. Another, albeit more unlikely,
explanation, is that the Granger causality test may not be suitable for anomaly
detection alerts: in fact, in [9] it has been tested on misuse detection alerts. But
in fact there are also theoretical reasons to doubt that the application of the
Granger test can lead to stable, good results. First, the test is asymptotic w.r.t.
p meaning that the results reliability decreases as p increases because of the loss
of degrees of freedom. Second, it is based on the strong assumption of linearity
in the auto-regressive model ﬁtting step, which strongly depends on the observed
phenomenon. In the same way, the stationarity assumption of the model does
not always hold.
5 Modeling alerts as stochastic processes
Instead of interpreting alert streams as time series (as proposed by the GCT-
based approach), we propose to change point of view by using a stochastic model
6.
in which alerts are modeled as (random) events in time. This proposal can be
seen as a formalized extension of the approach introduced in [1], which correlates
alerts if they are ﬁred by diﬀerent IDS within a “negligible” time frame, where
“negligible” is deﬁned with a crisp, ﬁxed threshold.
For simplicity, once again we describe our technique in the simple case of a
single HIDS and a single NIDS which monitors the whole network. The concepts,
however, can be easily generalized to take into account more than two alert
streams, by evaluating them couple by couple. For each alert, we have three
essential information: a timestamp, a “target” host (ﬁxed, in the case of the
HIDS, to the host itself), and the generating sensor (in our case, a binary value).
We reuse the scenario and data we already presented in Section 4 above.
With a self-explaining notation, we also deﬁne the following random variables:
TN etP are the arrival times of network alerts in N etP (TN etO , THostP are simi-
larly deﬁned); εN etP (εN etO ) are the delays (caused by transmission, processing
and diﬀerent granularity in detection) between a speciﬁc network-based alert
regarding pascal (not pascal) and the corresponding host-based one. The ac-
tual values of each T(·) is nothing but the set of timestamps extracted from the
corresponding alert stream. We reasonably assume that εN etP and TN etP are
stochastically independent (the same is assumed for εN etO and TN etO ).
In an ideal correlation framework with two equally perfect IDS with a 100%
DR and 0% FPR, if two alert streams are correlated (i.e., they represent inde-
pendent detections of the same attack occurrences by diﬀerent IDSes [1]), they
also are “close” in time. N etP and HostP should evidently be an example of
such a couple of streams. Obviously, in the real world, some alerts will be missing
(because of false negatives, or simply because some of the attacks are detectable
only by a speciﬁc type of detector), and the distances between related alerts will
therefore have some higher variability. In order to account for this, we can “cut
oﬀ” alerts that are too far away from a corresponding alert in the other time
series, presuming them to be singletons. In our case, knowing that single attacks
did not last more than 400s in the original dataset, we tentatively set a cutoﬀ
threshold at this point.
Under the given working assumptions and the proposed stochastic model, we
can formalize the correlation problem as a set of two statistical hypothesis tests:
H0 : THostP = TN etP + εN etP vs. H1 : THostP = TN etP + εN etP (1)
H0 : THostP = TN etO + εN etO vs. H1 : THostP = TN etO + εN etO (2)
Let {ti,k } be the observed timestamps of Ti ∀i ∈ {HostP, N etP, N etO}, the
meaning of the ﬁrst test is straightforward: within a random amount of time,
εN etP , the occurring of a host alert, tHostP,k , is preceded by a network alert,
tN etP,k . If this does not happen for a statistically signiﬁcant amount of events,
the test result is that alert stream TN etP is uncorrelated to THostP ; in this case,
we have enough statistical evidence for refusing H1 and accepting the null one.
Symmetrically, refusing the null hypothesis of the second test means that the
N etO alert stream (regarding to all hosts but pascal) is correlated to the alert
stream regarding pascal.
7.
HostP NetP HostP NetO
350
0.005
0.005
300
300
0.004
0.004
250
0.003
0.003
e_NetO
e_NetP
200
200
Density
Density
0.002
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150
100
100
0.001
0.001
50
0.000
0.000
0
0 100 200 300 400 100 200 300 400 0 100 200 300 400 100 200 300 400
e_NetP Estimator e_NetO Estimator
Fig. 4. Histograms vs. est. density (red dashes) and Q-Q plots, for both fˆ and fˆ .
O P
Note that, the above two tests are strongly related to each other: in an
ideal correlation framework, it cannot happen that both “N etP is correlated to
HostP ” and “N etO is correlated to HostP ”: this would imply that the network
activity regarding to all hosts but pascal (which raises N etO) has to do with the
host activity of pascal (which raises HostP ) with the same order of magnitude
of N etP , that is an intuitively contradictory conclusion. Therefore, the second
test acts as a sort of “robustness” criterion.
From our alerts, we can compute a sample of εN etP by simply picking, for
each value in N etP , the value in HostP which is closest, but greater (applying
a threshold as deﬁned above). We can do the same for εN etO , using the alerts in
N etO and HostP .
The next step involves the choice of the distributions of the random variables
we deﬁned above. Typical distributions used for modeling random occurrences
of timed events fall into the family of exponential Probability Density Functions
(PDF)s [20]. In particular, we decided to ﬁt them with Gamma PDFs, because
our experiments show that such a distribution is a good choice for both the
εN etP and εN etO .
The estimation of the PDF of εN etP , fP := fεN etP , and εN etO , fO :=
fεN etO , is performed using the well known Maximum Likelihood (ML) tech-
nique [21] as implemented in the GNU R software package: the results are sum-
marized in Fig. 4. fP and fO are approximated by Gamma[3.0606, 0.0178] and
Gamma[1.6301, 0.0105], respectively (standard errors on parameters are 0.7080,
0.0045 for fP and 0.1288, 0.009 for fO ). From now on, the estimator of a given
ˆ
density f will be indicated as f .
Fig. 4 shows histograms vs. estimated density (red, dashed line) and quantile-
ˆ ˆ
quantile plots (Q-Q plots), for both fO and fP . We recall that Q-Q plots are
an intuitive graphical “tool” for comparing data distributions by plotting the
quantile of the ﬁrst distribution against the quantile of the other one.
Considering that the samples sizes of ε(·) are around 40, Q-Q plots empirically
ˆ ˆ
conﬁrms our intuition: in fact, fO and fP are both able to explain real data
ˆ ˆ
well, within inevitable but negligible estimation errors. Even if fP and fO are
both Gamma-shaped, it must be noticed that they signiﬁcantly diﬀer in their
parametrization; this is a very important result since it allows to set up a proper
criterion to decide whether or not εN etP and εN etO are generated by the same
phenomenon.
8.
HostP − NetP HostP − NetO
0.06
0.06
0.04
0.04
Density
Density
0.02
0.02
0.00
0.00
10 20 30 40 50 0 10 20 30 40 50 60
e_NetP e_NetO
Fig. 5. Histograms vs. est. density (red dashes) for both fˆ and fˆ (IDEVAL 1998)
O P
Given the above estimators, a more precise and robust hypotheses test can
be now designed. The Test 1 and 2 can be mapped into two-sided Kolmogorov-
Smirnov (KS) tests [20], achieving the same result in terms of decisions:
H0 : εN etP ∼ fP vs. H1 : εN etP ∼ fP (3)
H0 : εN etO ∼ fO vs. H1 : εN etO ∼ fO (4)
where the symbol ∼ means “has the same distribution of”. Since we do not
know the real PDFs, estimators are used in their stead. We recall that the KS-
test is a non-parametric test to compare a sample (or a PDF) against a PDF (or
a sample) to check how much they diﬀers from each other (or how much they
ﬁt). Such tests can be performed, for instance, with ks.test() (a GNU R native
procedure): resulting p-values on IDEVAL 1999 are 0.83 and 0.03, respectively.
Noticeably, there is a signiﬁcant statistical evidence to accept the null hy-
pothesis of Test 3. It seems that the ML estimation is capable of correctly ﬁtting
a Gamma PDF for fP (given εN etP samples), which double-checks our intuition
about the distribution. The same does not hold for fO : in fact, it cannot be
correctly estimated, with a Gamma PDF, from εN etO . The low p-value for Test
4 conﬁrms that the distribution of εN etO delays is completely diﬀerent than the
one of εN etP . Therefore, our criterion doest not only recognize noisy delay-based
relationships among alerts stream if they exists; it is also capable of detecting if
such a correlation does not hold.
We also tested our technique on alerts generated by our NIDS/HIDS running
on IDEVAL 1998 (limiting our analysis to the ﬁrst four days of the ﬁrst week),
in order to cross-validate the above results. We prepared and processed the
data with the same procedures we described above for the 1999 dataset. Start-
ing from almost the same proportion of host/net alerts against either pascal
or other hosts, the ML-estimation has computed the two Gamma densities
shown in Fig. 5: fP and fO are approximated by Gamma(3.5127, 0.1478) and
Gamma(1.3747, 0.0618), respectively (standard errors on estimated parameters
are 1.3173, 0.0596 for fP and 0.1265, 0.0068 for fO ). These parameter are very
similar to the ones we estimated for the IDEVAL 1999 dataset. Furthermore,
with p-values of 0.51 and 0.09, the two KS tests conﬁrm the same statistical
discrepancies we observed on the 1999 dataset.
The above numerical results show that, by interpreting alert streams as ran-
dom processes, there are several (stochastic) dissimilarities between net-to-host
9.
delays belonging to the same net-host attack session, and net-to-host delays be-
longing to diﬀerent sessions. Exploiting these dissimilarities, we may ﬁnd out
the correlation among streams in an unsupervised manner, without the need to
predeﬁne any parameter.
6 Conclusions
In this paper we analyzed the use of of diﬀerent types of statistical tests for the
correlation of anomaly detection alerts, a problem which has little or no solu-
tions available today. One of the few correlation proposals that can be applied
to anomaly detection is the use of a Granger Causality Test (GCT). After dis-
cussing a possible testing methodology, we observed that the IDEVAL datasets
traditionally used for evaluation have various shortcomings, that we partially
addressed by using the data for a simpler scenario of correlation, investigating
only the link between a stream of host-based alerts for a speciﬁc host, and the
corresponding stream of alerts from a network based detector.
We examined the usage of a GCT as proposed in earlier works, showing that
it relies on the choice of non-obvious conﬁguration parameters which signiﬁcantly
aﬀect the ﬁnal result. We also showed that one of these parameters (the order
of the models) is absolutely critical, but cannot be uniquely estimated for a
given system. Instead of the GCT, we proposed a simpler statistical model of
alert generation, describing alert streams and timestamps as stochastic variables,
and showed that statistical tests can be used to create a reasonable criterion for
distinguishing correlated and non correlated streams. We proved that our criteria
work well on the simpliﬁed correlation task we used for testing, without requiring
complex conﬁguration parameters.
This is an exploratory work, and further investigations of this approach on
real, longer sequences of data, as well as further reﬁnements of the tests and the
criteria we proposed, are surely needed. Another possible extension of this work
is the investigation of how these criteria can be used to correlate anomaly and
misuse-based alerts together, in order to bridge the gap between the existing
paradigms of intrusion detection.
Acknowledgments
We need to thank prof. Ilenia Epifani and prof. Giuseppe Serazzi for their helpful
comments, as well as the anonymous reviewers.
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